Finite De Finetti Theorem Achieves Convergent Polynomial Optimization for Complex Bodies

Scientists have long sought to understand the limits of approximation in complex optimisation problems, and a new finite de Finetti representation theorem offers a significant step forward. Julius A. Zeiss, Gereon Koßmann (both from the Institute for Quantum Information, RWTH Aachen University), and René Schwonnek, Martin Plávala (from Leibniz Universit at Hannover) demonstrate this theorem for general convex bodies, utilising a novel approach based on relative entropy in general probabilistic theories. This breakthrough addresses a core challenge in polynomial optimisation , guaranteeing the convergence of outer hierarchies, particularly for problems featuring inequality constraints , and establishes a uniform bound on mutual information applicable to diverse convex bodies. By generalising a quantitative monogamy argument, the team not only proves the theorem but also delivers a constructive rounding scheme for certified solutions, with implications extending to the approximation of optimal values in non-local games.

The research team achieved this by generalizing a quantitative monogamy-of-entanglement argument from quantum theory to apply to arbitrary convex bodies, establishing a uniform upper bound on mutual information in multipartite extensions. This innovative approach leads to a finite de Finetti theorem and, consequently, a convergent conic hierarchy applicable to a broad class of polynomial optimization problems encompassing both equality and inequality constraints.

Experiments show the team developed a constructive rounding scheme that yields certified interior points with controlled approximation error, a significant advancement in the field. As an application of their techniques, they successfully expressed the optimal GPT value of a two-player non-local game as a polynomial optimization problem, enabling the creation of approximation schemes with finite convergence guarantees. The core of this work lies in an information-theoretic finite de Finetti representation theorem, informally stating that permutation-invariant states on a B-system relative to system A converge to a separable state with a bounded norm, specifically ∥ xAB −xAB∥≤c(A, B) √n. This theorem, building upon existing de Finetti theorems in quantum theory, provides a quantitative approximation bound for general convex bodies, unlike previous qualitative results.
The study unveils a crucial insight: entanglement in GPTs is inherently limited in its shareability, and the de Finetti result provides a precise quantitative expression of this monogamy. This achievement is a consequence of several key ideas, beginning with an integral representation of relative entropy extended to GPTs, allowing for the definition of mutual information in this broader context. Within this framework, the researchers proved that mutual information admits a uniform upper bound under extensions of one subsystem, establishing that I(A : B)xAB ≤λA(1 + ln λA), where λA is a constant dependent only on system A. This bound, I(A : Bn 1 )xABn 1 ≤c(A), demonstrates that shared information between A and any other party B is uniformly bounded, solidifying a strong monogamy-of-entanglement relation.

Furthermore, the research establishes a convergent conic hierarchy for polynomial optimization over convex bodies, resolving an open problem regarding the accommodation of inequality constraints, which neither the DPS nor the polarization hierarchy previously guaranteed. This novel hierarchy incorporates both local equality and inequality constraints, yielding explicit, finite-level convergence guarantees, a major improvement over existing methods. The team also provides a certified rounding procedure, generating interior approximations with explicitly defined accuracy bounds, as detailed in Table 1, which compares their work to the polarization hierarchy, highlighting the quantified finite-level guarantees and certified construction of interior points.

De Finetti Theorem and Conic Hierarchy Development represent

Scientists recently developed a finite de Finetti representation theorem for general convex bodies, leveraging a new notion of relative entropy in general probabilistic theories (GPTs). The research team engineered a method to address a long-standing question in polynomial optimization, the existence of a convergent outer hierarchy for problems featuring inequality constraints and analytical convergence guarantees. This strategy generalises a quantitative monogamy argument, establishing a uniform upper bound on mutual information in multipartite extensions, ultimately leading to a finite de Finetti theorem and a convergent conic hierarchy applicable to a broad range of polynomial optimization problems. To achieve this, researchers first extended a recently introduced integral representation of relative entropy to GPTs, enabling the definition of mutual information within this broader framework.

This innovative approach overcomes the lack of a canonical definition of mutual information in general conic settings, previously well-understood only within quantum theory utilising Umegaki’s relative entropy. Experiments employed Proposition 3.5, demonstrating that mutual information admits a uniform upper bound under extensions of one subsystem: I(A : B)xAB ≤ λA(1 + ln λA), where λA is a constant dependent solely on system A. Consequently, the study revealed I(A : Bn 1 )xABn 1 ≤ c(A), establishing a strong monogamy-of-entanglement relation independent of the number of extensions, n. The team then pioneered a constructive rounding scheme, yielding certified interior points with controlled approximation error, quantified by ∥ xAB −xAB∥≤c(A, B) √n.

This technique reveals that n-extendible states in a maximal tensor product space converge to states in a minimal tensor product space, a consequence of the established information-theoretic bounds. Furthermore, the work addresses a critical gap in existing methods by constructing a convergent conic hierarchy incorporating both local equality and inequality constraints, delivering explicit, finite-level convergence guarantees, a significant improvement over asymptotic guarantees offered by methods like the DPS or polarization hierarchy. Table 1 summarises these advancements, highlighting the quantified finite-level guarantees and certified construction of interior points achieved in this research.

Finite de Finetti Theorem and Conic Hierarchy Convergence

Scientists have established a finite de Finetti representation theorem applicable to general convex bodies, leveraging a recently proposed notion of relative entropy in general probabilistic theories. This breakthrough addresses a long-standing challenge in polynomial optimization, achieving convergent outer hierarchies with analytical guarantees, particularly for problems incorporating both equality and inequality constraints. Experiments revealed a uniform upper bound on mutual information in multipartite extensions, a crucial step towards a finite de Finetti theorem and, consequently, a convergent conic hierarchy. The team measured explicit, finite-level convergence, overcoming a major limitation of previous methods and enabling the construction of a conic hierarchy incorporating local equality and inequality constraints.

Results demonstrate a constructive rounding scheme yielding certified interior points with controlled approximation error; tests prove the accuracy of these points is explicitly bounded by the de Finetti theorem. Measurements confirm the ability to obtain matching upper and lower bounds, effectively sandwiching the original problem value and providing a robust solution. Data shows a novel link between information-theoretic de Finetti arguments and the theory of conic optimization, resolving shortcomings of prior methods. The research successfully reformulated the problem of determining the optimal value of a two-player non-local game as a polynomial optimization problem, allowing for the development of approximation schemes with finite convergence guarantees.

Scientists recorded that if underlying convex bodies admit suitable conic lifts, the hierarchy can be implemented at the lifted cone level, enhancing computational efficiency. The breakthrough delivers analytical finite-level convergence guarantees, ensuring constraint compatibility beyond the averaged sense inherent in previous approaches. Specifically, the work addresses an open problem by incorporating inequality constraints into the de Finetti-based convergence framework, bridging the gap between de Finetti-type methods and the Lasserre and Sherali-Adams hierarchies. Measurements.